Kurdyka- Lojasiewicz inequality and Kurdyka- Lojasiewicz - PowerPoint PPT Presentation

Kurdyka-� Lojasiewicz inequality and Kurdyka- � Lojasiewicz inequality and subgradient trajectories : the convex case subgradient trajectories: the convex case Olivier Ley Olivier Ley Journ´ ees Universit´ e de Tours Franco- Chiliennes www.lmpt.univ-tours.fr/ ∼ ley Toulon, April 2008 Joint work with : J´ erˆ ome Bolte (Paris vi ) Aris Daniilidis (U. Autonoma Barcelona & Tours) and Laurent Mazet (Paris xii )

Lojasiewicz inequality � Kurdyka- Lojasiewicz � inequality and subgradient Lojasiewicz inequality [� � Lojasiewicz 1963] trajectories: the convex f : R N → R is analytic . case Let a be a critical point of f . Olivier Ley Then there exists a neighborhood U of a , C > 0 and θ ∈ (0 , 1) Journ´ ees Franco- such that Chiliennes Toulon, April 2008 ||∇ f ( x ) || ≥ | f ( x ) − f ( a ) | θ for all x ∈ U . ➪ Finite length of the gradient trajectories, Every critical point is limit of a gradient trajectory, etc.

Basic assumptions Kurdyka- � Lojasiewicz To simplify : inequality and subgradient trajectories: the convex H is a real Hilbert space case f : H → [0 , + ∞ ) is smooth ( f ≥ 0) Olivier Ley For all r > 0 , C r := { f ≤ r } Journ´ ees Franco- Chiliennes (A0) (0 is a critical point and a global minimum) Toulon, April 2008 0 ∈ C 0 (A1) (0 is an isolated critical value) There exits r 0 > 0 such that : x ∈ C r 0 and f ( x ) > 0 ⇒ ∇ f ( x ) � = 0 (A2) (Sublevel compactness) There exits r 0 > such that : C r 0 = { f ≤ r 0 } is compact.

Generalization : K� L-inequality Kurdyka- Lojasiewicz � inequality and We say that f satisfies Kurdyka-� Lojasiewicz inequality subgradient trajectories: [Kurdyka 1998] if : the convex case There exists ϕ ∈ KL (0 , r 0 ) such that : Olivier Ley Journ´ ees ||∇ ( ϕ ◦ f )( x ) || ≥ 1 for all x ∈ C r 0 \ C 0 . Franco- Chiliennes Toulon, April 2008 where : � KL (0 , r 0 ) = ϕ : [0 , r 0 ] → R + continuous , ϕ (0) = 0 , ϕ ∈ C 1 (0 , r 0 ) , ϕ ′ > 0 � . ◮ Lojasiewicz inequality is a particular case with C (1 − θ ) r 1 − θ . 1 ϕ ( r ) =

The convex case Kurdyka- Lojasiewicz � inequality and subgradient trajectories: the convex case From now on, we assume that Olivier Ley Journ´ ees f is convex Franco- Chiliennes Toulon, April Issues : 2008 1 Characterizations of the K� L-inequality in the convex case 2 Does a convex function satisfy the K� L-inequality ?

Piecewise gradient curves Gradient dynamical system : Kurdyka- Lojasiewicz � � ˙ inequality and X x ( t ) = −∇ f ( X x ( t )) , t ≥ 0 subgradient trajectories: X x (0) = x the convex case A piecewise gradient curve γ is a countable family of Olivier Ley gradient curves X x i ([0 , t i )) with Journ´ ees Franco- f ( X x i (0)) = f ( x i ) = r i , f ( X x i ( t i )) = r i +1 r i ↓ 0 Chiliennes Toulon, April i → + ∞ x 0 2008 X x 0 ( t ) C r 0 x 1 C r 2 C r 1 x 2 C r 3 C 0 0

Classical properties of the ‘convex’ gradient curves Kurdyka- Lojasiewicz � inequality and Lemma. For all x 0 ∈ C r 0 \ C 0 , subgradient trajectories: 1 t �→ f ( X x 0 ( t )) is convex, L 1 (0 , + ∞ ) and decreasing with the convex case limit 0 . Olivier Ley 2 Each trajectory goes closer to all minima at the same time, Journ´ ees i.e., for each a ∈ C 0 , Franco- Chiliennes Toulon, April d 2008 dt || X x 0 ( t ) − a || 2 ≤ − 2 f ( X x 0 ( t )) < 0 . 3 For all T > 0 , � T 1 || ˙ � X x 0 ( t ) || dt ≤ √ || x 0 || log T . 2 0

Characterizations of the K� L-inequality ( f convex ) Kurdyka- Lojasiewicz � inequality and Theorem. The following statements are equivalent : subgradient trajectories: the convex 1 f satisfies the K� L-inequality in C r 0 : case ||∇ ( ϕ ◦ f )( x ) || ≥ 1 with ϕ ∈ KL (0 , r 0 ) . Olivier Ley 2 f satisfies the K� L-inequality globally in H with Journ´ ees Franco- ϕ ∈ KL (0 , + ∞ ) which is concave . Chiliennes Toulon, April 1 3 r ∈ (0 , r 0 ] �→ f ( x )= r ||∇ f ( x ) || is integrable. 2008 inf 4 For all piecewise gradient curves γ in C r 0 we have � t i ∞ � || ˙ length ( γ ) = X x i ( t ) || dt < ∞ . 0 i =0

Length of the ‘convex’ gradient curves Kurdyka- Lojasiewicz � ezis 1973] Given x 0 ∈ C r 0 , do we have inequality and [Br´ subgradient trajectories: � ∞ the convex || ˙ case length ( X x 0 ) = X x 0 ( t ) || dt < ∞ ? Olivier Ley 0 Journ´ ees Theorem. [Br´ ezis 1973] Yes if int(argmin( f )) � = ∅ . Franco- Chiliennes Toulon, April finite length finite length 2008 X x 0 ( t ) ˙ X x 0 ( t ) C 0 C 0 unknown length Theorem. [Baillon 1978] No in general (counter-example in infinite dimension)

A sufficient condition for a convex function to satisfy K� L-inequality Kurdyka- � Lojasiewicz inequality and Theorem. Assume that there exists subgradient trajectories: m : [0 , + ∞ ) → [0 , + ∞ ) continuous increasing with m (0) = 0 the convex case such that f ≥ m ( dist ( · , C 0 )) on C r 0 and Olivier Ley � r 0 m − 1 ( r ) Journ´ ees Franco- dr < + ∞ ( growth condition ) . (1) Chiliennes r 0 Toulon, April 2008 Then K� L-inequality holds for f . ◮ Proof : f ( x ) ≤ �∇ f ( x ) , x − p C 0 ( x ) � ≤ ||∇ f ( x ) || dist ( x , C 0 ) ≤ ||∇ f ( x ) || m − 1 ( f ( x )) . ◮ non analytic examples : m ( r ) = exp ( − 1 / r α ) , α ∈ (0 , 1) satisfies (1).

A smooth convex counterexample to K� L-inequality Kurdyka- Lojasiewicz � inequality and subgradient trajectories: the convex case Olivier Ley Theorem. There exists a C 2 convex function f : R 2 → R + Journ´ ees Franco- with { f = 0 } = D (0 , 1) for which K� L-inequality fails. Chiliennes Toulon, April 2008 ◮ Note that the gradient trajectories have uniform finite length.

An auxiliary problem Kurdyka- Lojasiewicz � A farmer rakes its (convex) field in several steps in the following inequality and subgradient way : trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April ℓ 0 ℓ 1 2008 ℓ 2 If he is unlucky, is it possible that he walks an infinite path ? � (i.e. ℓ i = + ∞ ) i ≥ 0

An auxiliary problem Kurdyka- Lojasiewicz � A farmer rakes its (convex) field in several steps in the following inequality and subgradient way : trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April ℓ 0 ℓ 1 2008 ℓ 2 If he is unlucky, is it possible that he walks an infinite path ? � (i.e. ℓ i = + ∞ ) i ≥ 0 Answer : Yes !

Hausdorff distance between nested convex sets Kurdyka- � Lojasiewicz inequality and subgradient trajectories: Lemma. There exists a decreasing sequence of compact convex the convex subsets { T k } k in R 2 such that : case Olivier Ley ( i ) T 0 is the disk D := D (0 , 2) ; Journ´ ees Franco- ( ii ) T k +1 ⊂ int T k for every k ∈ N ; Chiliennes Toulon, April � 2008 ( iii ) T k is the unit disk D (0 , 1) ; k ∈ N + ∞ � ( iv ) dist Hausdorff ( T k , T k +1 ) = + ∞ . k =0

Picture of the sequence of convex sets Kurdyka- Lojasiewicz � inequality and subgradient trajectories: the convex case Olivier Ley Journ´ ees Franco- Chiliennes Toulon, April 2008 1 √ ℓ k := dist Hausdorff ( T k , T k +1 ) ≈ R i − R i +1 and N i ≈ R i − R i +1 It suffices to take R i − R i +1 = 1 i 2 in order that � i R i − R i +1 < ∞ and � � k ℓ k ≈ � i N i ( R i − R i +1 ) ≈ � R i − R i +1 = + ∞ . i

End of the construction Kurdyka- Lojasiewicz � inequality and subgradient trajectories: the convex case Olivier Ley ◮ Construction of a convex function with prescribed sublevel Journ´ ees sets T k : [Torralba 1996]. Franco- Chiliennes Toulon, April 2008 ◮ Smoothing of the function.